We make a second modification from $(\tilde{g}^{(3)},\tilde{\omega}^{(3)},\Omega)$ to another new Kähler ansatz $(\tilde{g}^{(4)},\tilde{\omega}^{(4)},\Omega)$ designed to match up with the Taub-NUT type metric $(g_{\mathbb{C}^{3}},\omega_{\mathbb{C}^{3}},\Omega_{\mathbb{C}^{3}})$ in Chapter 2.

Recall from Theorem 2.26 that there is a Kähler potential $\phi^{\mathbb{C}^{3}}$ such that

$$\omega_{\mathbb{C}^{3}}=\omega^{(2)}+\sqrt{-1}\partial\bar{\partial}\phi^{\mathbb{C}^{3}}=\omega^{(1)}+\sqrt{-1}\partial\bar{\partial}\phi^{\mathbb{C}^{3}},\quad\text{for }|\vec{\mu}|_{a}\gtrsim A^{-1/4},$$

with bound $\left\lVert d\phi^{\mathbb{C}^{3}}\right\rVert_{C^{k+1,\alpha}_{-\epsilon,-1+\epsilon}(\mathbb{C}^{3})}\leq CA^{-1/4}.$ We can impose a normalisation such that $|\phi^{\mathbb{C}^{3}}|\leq CA^{-1/2+\frac{3}{4}\epsilon}$ for $\frac{1}{100}A^{1/2}\lesssim|\vec{\mu}|_{a}\leq\frac{1}{3}A^{1/2}.$

We then define a modified Kähler metric ansatz $\tilde{\omega}^{(4)}$ on $M^{+}_{\nu}$. Take a standard cutoff function

$$\chi(s)=\begin{cases}1\quad s\leq 1,\\ 0\quad s\geq 2,\end{cases}$$

and define

$$\tilde{\omega}^{(4)}=\tilde{\omega}^{(3)}+\sqrt{-1}\partial\bar{\partial}\phi_{4},\quad\phi_{4}=\chi(\frac{|\vec{\mu}|_{a}}{\frac{1}{12}A^{1/2}})\phi^{\mathbb{C}^{3}}-2\varphi_{3}.$$

In particular

$$\begin{cases}\tilde{\omega}^{(4)}=\omega_{\mathbb{C}^{3}}-2\sqrt{-1}\partial\bar{\partial}\varphi_{3},\quad&|\vec{\mu}|_{a}\leq\frac{1}{12}A^{1/2},\\ \tilde{\omega}^{(4)}=\tilde{\omega}^{(3)},\quad&|\vec{\mu}|_{a}\geq\frac{1}{3}A^{1/2}.\end{cases}$$

The positive definiteness of $\tilde{\omega}^{(4)}$ follows from the metric deviation estimate:

$$\begin{cases}\left\lVert\partial\bar{\partial}\varphi_{3}\right\rVert_{C^{k,\alpha}_{0,0}(\mathbb{C}^{3}\cap\{|\vec{\mu}|_{a}\leq\frac{1}{3}A^{1/2}\})}\leq C\nu A^{-3/4},\\ \left\lVert\partial\bar{\partial}\{\chi(\frac{|\vec{\mu}|_{a}}{\frac{1}{12}A^{1/2}})\phi^{\mathbb{C}^{3}}\}\right\rVert_{C^{k,\alpha}_{-1-\epsilon,0}(|\vec{\mu}|_{a}\sim A^{1/2})}\leq CA^{3/4(-1+\epsilon)}.\end{cases}$$

Here $\varphi_{3}$ is inserted to approximately cancel the cutoff error $\Delta_{a}\varphi_{3}$ in the volume form error $\tilde{E}^{(3)}$ (cf. Lemma 3.28).